International Mathematical Olympiad 1997 Problem 4

An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots, 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots, n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no silver matrix for $n = 1997$;

(b) silver matrices exist for infinitely many values of $n$.